Advances in Differential Equations

Renormalized entropy solutions of homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations

Yachun Li, Qin Wang, and Zhigang Wang

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Abstract

We study the well-posedness of renormalized entropy solutions for quasilinear anisotropic degenerate parabolic-hyperbolic equation of the type $\partial_{t}u+\text{div}f(u)=\nabla\cdot(a(u)\nabla u)$ in a bounded domain with general $L^{1}$ initial data and homogeneous Dirichlet boundary condition. We use the device of doubling variables to prove the uniqueness and use the vanishing viscosity method to prove the existence.

Article information

Source
Adv. Differential Equations Volume 19, Number 3/4 (2014), 387-408.

Dates
First available in Project Euclid: 30 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.ade/1391109090

Mathematical Reviews number (MathSciNet)
MR3161666

Subjects
Primary: 35K20: Initial-boundary value problems for second-order parabolic equations 35K59: Quasilinear parabolic equations 35K65: Degenerate parabolic equations 35L04: Initial-boundary value problems for first-order hyperbolic equations 35L65: Conservation laws 35M13: Initial-boundary value problems for equations of mixed type 35Q35: PDEs in connection with fluid mechanics 76R99: None of the above, but in this section 76S05: Flows in porous media; filtration; seepage

Citation

Li, Yachun; Wang, Qin; Wang, Zhigang. Renormalized entropy solutions of homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations. Adv. Differential Equations 19 (2014), no. 3/4, 387--408. https://projecteuclid.org/euclid.ade/1391109090.


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